The crucial damage was done in the first half of the second leg after a wonderful curling effort from Dynamo's Andriy Yarmolenko in the 21st minute had been equalised in similar fashion eight minutes later by Romelu Lukaku.

Six minutes later, Everton's slow reactions at the back allowed Lukasz Teodorczyk to prod home, and within another two minutes, Miguel Veloso met a failed attempted clearance by Toffees captain Phil Jagielka with a volley that deflected in.

Roberto Martinez's men then once again did not cover themselves in glory defensively 11 minutes into the second half when Dynamo added another goal, Oleg Gusev finishing from close-range on the rebound.

And the triumph for the Ukrainian league leaders was wrapped up in the 76th minute when Vitorino Antunes let fly superbly from distance, before Jagielka headed in an 82nd-minute consolation.

Everton had been hoping to make a European quarter final for the first time in 30 years, but instead completed a treble of eliminations for the Premier League's last three remaining sides in Europe over the past few days following Arsenal and Manchester City's Champions League exits.

While the questions about the real strength of English football will only continue from here, Martinez's players, their creditable run in this competition now over, will focus on trying to finish their disappointing Premier League season positively, with a trip to QPR next up on Sunday as they try to build on the morale-boosting 3-0 win over Newcastle.

Everton always looked set to have plenty to deal with in the shape of a Dynamo outfit with a formidable home record, as well as the poor pitch and hostile atmosphere of the Olympic Stadium, a section of which had been closed for this match by UEFA after crowd trouble at the ground in the last round.

And the pressure was on the visitors from the off, Gusev looping a header over and Antunes firing wide in the early stages.

After Ross Barkley replied with a shot gathered by Dynamo goalkeeper Olexandr Shovkovskiy, the home team grabbed the lead in emphatic style as Yarmolenko burst forward, bamboozled Gareth Barry with his footwork and sent a fine strike past Tim Howard from outside the box.

Everton swiftly rallied as Barkley sent a low drive against the far post, before a decent Howard save kept the lively Yarmolenko at bay.

The Toffees fans who had travelled for the game - estimated to be just under 1,000 in number - were then sent into raptures as Lukaku, his club's record goalscorer in Europe, took his tally to eight in the competition.

The striker collected the ball when it came his way via a blocked Christian Atsu shot, rode one challenge and beat Shovkovskiy beautifully from the edge of the area.

Only a last-ditch tackle then prevented Steven Naismith from putting Everton ahead on the night, but moments later the Toffees shot themselves in the foot as they failed to deal with a cross into the box and Yarmolenko teed up Teodorczyk for an easy finish.

The Merseysiders were still digesting that when they conceded another soft goal, Jagielka only managing to divert a cross to Veloso, whose volley went in off Seamus Coleman.

Everton needed something dramatic to happen in their favour if they were to have any chance of salvaging the tie, but another defensive lapse saw them fall further behind as a low cross came into their box and Gusev was able to shoot against Coleman, then score on the follow-up.

Lukaku had a shot well saved by Shovkovskiy and Barkley saw another strike come back off the woodwork, but Dynamo were soon on the attack again, and after Howard had denied Teodorczyk, Antunes cracked in perhaps the best goal of the night.

Dynamo kept driving forward, with Yarmolenko rattling the bar, before Everton fans finally had something - albeit something relatively meaningless - to cheer again as Jagielka nodded in from Leighton Baines' corner.